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ANALYTICAL APPROXIMATIONS TO HYDROSTATIC SOLUTIONS AND
SCALING LAWS OF CORONAL LOOPS
Markus J. Aschwanden and Carolus J. Schrijver Lockheed Martin Advanced Technology Center, Solar and Astrophysics Laboratory, Department L9-41, Building 252, 3251 Hanover Street, Palo Alto, CA 94304; aschwanden@lmsal.com Received 2002 February 14; accepted 2002 May 22
ABSTRACT We derive accurate analytical approximations to hydrostatic solutions of coronal loop atmospheres, applicable to uniform and nonuniform heating in a large parameter space. The hydrostatic solutions of the temperature TðsÞ, density neðsÞ, and pressure profile pðsÞ as a function of the loop coordinate s are explicitly expressed in terms of three independent parameters: the loop half-length L, the heating scale length sH , and either the loop-top temperature Tmax or the base heating rate EH 0. The analytical functions match the numer- ical solutions with a relative accuracy of d 10 ^2 –10^3. The absolute accuracy of the scaling laws for loop base pressure p 0 (L, sH , Tmax) and base heating rate EH 0 (L, sH , Tmax), previously derived for uniform heating by Rosner et al., and for nonuniform heating by Serio et al., is improved to a level of a few percent. We generalize also our analytical approximations for tilted loop planes (equivalent to reduced surface gravity) and for loops with varying cross sections. There are many applications for such analytical approximations: (1) the improved scaling laws speed up the convergence of numeric hydrostatic codes as they start from better initial values, (2) the multitemperature structure of coronal loops can be modeled with multithread concepts, (3) line-of-sight integrated fluxes in the inhomogeneous corona can be modeled with proper correction of the hydrostatic weighting bias, (4) the coronal heating function can be determined by forward-fitting of soft X-ray and EUV fluxes, or (5) global differential emission measure distributions dEM=dT of solar and stellar coronae can be simulated for a variety of heating functions. Subject headings: hydrodynamics — stars: coronae — Sun: corona
- INTRODUCTION Accurate density and temperature models are funda- mental tools to explore the physical processes of plasma heating and cooling in solar and stellar coronae. The tre- mendous increase of imaging data in soft X-rays and extreme ultraviolet (EUV) produced by the Yohkoh, SoHO, TRACE, ROSAT, ASCA, Chandra, and Newton spacecraft have stimulated modeling efforts in an unpre- cedented way. The modeling of coronal loops with hydrostatic solutions, which ensure the basic physical conservation laws of mass, momentum, and energy, how- ever, is computation-expensive with numeric codes, par- ticularly for large sets to model entire stellar coronae. Therefore, appropriate analytical expressions for their density and temperature profiles are highly desirable. At this time, no analytical solutions are known for the hydrostatic equations, except for an approximate temper- ature function in the special case of uniform heating, constant cross section, and zero gravity (Rosner, Tucker, & Vaiana 1978; Kuin & Martens 1982). In this study we use a numerical code to compute some 1000 hydrostatic solutions in a large parameter space, for uniform as well as for nonuniform heating functions, and develop accu- rate analytical approximations by fitting them to the numerical solutions. By the same token, we quantify also the well-known scaling laws of loop base pressure and heating rate as derived earlier by Rosner et al. (1978) and Serio et al. (1981), but with higher precision, and add loop expansion as a parameter. The new analytical for- mulation consists of explicit expressions as a function of five independent parameters and can conveniently be applied to forward-fitting of coronal data and statistical
studies of solar and stellar atmospheres (e.g., Schrijver & Aschwanden 2002). The content of the paper includes a definition of the hydrodynamic equations as they are used here (x 2), a brief description of a numerical code that is used to calculate the exact hydrostatic solutions (x 3), the derivation of analytical approximations and more accurate scaling laws (x 4), gener- alizations for very short heating scale lengths, inclined loops, and loops expanding with height (x 5), and a discus- sion of applications (x 6).
- HYDRODYNAMIC EQUATIONS We define the quantities of the time-independent hydro- dynamic equations used in this study, which have been used with slightly different notations, approximations, assump- tions, and degree of completeness in previous work (e.g., Parker 1958; Rosner et al. 1978; Priest 1982; Mariska et al. 1982; Craig & McClymont 1986; Klimchuk, Antiochos, & Mariska 1987; Klimchuk & Mariska 1988; Withbroe 1988; Bray et al. 1991). The one-dimensional, time-independent (d=dt ¼ 0) hydrodynamic equations involve the equations of mass con- servation,
1 A
d ds
ðnvAÞ ¼ 0 ; ð 1 Þ
the momentum equation,
mnv
dv ds
dp ds
þ
dpgrav dr
dr ds
; ð 2 Þ
The Astrophysical Journal Supplement Series, 142:269–283, 2002 October
2002. The American Astronomical Society. All rights reserved. Printed in U.S.A.
and the energy equation (expressed in conservative form),
1 A
d ds
ðnvA½enth þ kin þ grav þ AFC Þ ¼ EH þ ER ; ð 3 Þ
where s is the distance along the loop measured from the solar surface, r is the radial distance to Sun center, AðsÞ is the loop cross section, m is the average particle mass, nðsÞ is the particle density, vðsÞ is the velocity of a single fluid, pðsÞ is the gas pressure, pgravðrÞ is the gravitational pressure, enthðsÞ is the enthalpy, kinðsÞ is the kinetic energy, gravðrÞ is the gravitational potential, FC ðsÞ is the conductive flux, EH ðsÞ is the volumetric heating rate, and ERðsÞ is the volu- metric radiative loss rate. The mass density (also called ¼ mn) of a fully ionized gas is defined by
mnðsÞ ¼ meneðsÞ þ miniðsÞ lmpneðsÞ ; ð 4 Þ
with mi ¼ lmp the average ion mass (i.e., l ð 10 1 þ 1 4 Þ= 11 ¼ 1 :3 for a coronal composition of H : He ¼ 10 : 1), mp ¼ 1 : 67 10 ^24 g the proton mass, and the density nðsÞ is assumed to be equal for electrons and ions (n ¼ ne ¼ ni) in a fully ionized gas. The total pressure pðsÞ of a fully ionized gas is defined by the ideal gas law and relates to the (electron) density nðsÞ by
pðsÞ ¼ ½neðsÞ þ niðsÞkBTðsÞ 2 nðsÞkBTðsÞ ; ð 5 Þ
where kB ¼ 1 : 38 10 ^16 erg K^1 is the Boltzmann constant and TðsÞ is the electron temperature. The enthalpy energy enthðsÞ comprises the heat energy acquired (or lost) at constant volume, plus the work done against the pressure force when the volume changes, and is defined by
enthðsÞ ¼ 52 kBTðsÞ ; ð 6 Þ
the kinetic energy kinðsÞ is
kinðsÞ ¼ 12 mv^2 ðsÞ ; ð 7 Þ
the gravitational potential gravðrÞ is
gravðrÞ ¼
GMm r
¼ mg
R^2
r
; ð 8 Þ
with the solar gravitation g ¼ GM=R^2 ¼ 2 : 74 104 cm s^2 and solar radius R ¼ 6 : 96 1010 cm. The differential gravitational pressure, used in the momentum equation (2), is
dpgrav dr
ðrÞ ¼
GMmn r^2
¼ mng
R^2
r^2
: ð 9 Þ
The next term of the energy balance equation describes the divergence of the conductive flux, which in a one-dimen- sional flux tube model is
FC ðsÞ ¼^ T^5 =^2 ðsÞ^
dTðsÞ ds
d ds
T^7 =^2 ðsÞ
; ð 10 Þ
with ¼ 9 : 2 10 ^7 erg s^1 cm^1 K7/2^ the Spitzer conduc- tivity. The most unknown term is the volumetric heating rate EH ðsÞ along the loop, which crucially depends on assump- tions on the physical heating mechanism. Many previous loop models assumed uniform heating, EH ðsÞ ¼ const (e.g.,
Rosner et al. 1978), for sake of simplicity. Here we parame- terize the heating function with two parameters: with the base heating rate EH 0 and an exponential scale length sH , as it was introduced by Serio et al. (1981),
EH ðsÞ ¼ E 0 exp
s sH
¼ EH 0 exp
s s 0 sH
: ð 11 Þ
While Serio’s base heating rate E 0 refers to the photosphere (at s ¼ 0), we introduce a base heating rate EH 0 that refers to the same reference height s ¼ s 0 as we will refer the base temperature T 0 , the base pressure p 0 , and the base density n 0. This Ansatz allows us to model nonuniform heating localized above the loop footpoints from arbitrary small heating scale lengths (sH 5 L) up to the limit of uniform heating (sH 4 L). Alternative parameterizations of heating functions that are suitable for loop-top heating have been used elsewhere (e.g., Priest et al. 2000; MacKay et al. 2000). The radiative losses ERðsÞ are proportional to the square of the electron density, n^2 e ðsÞ, multiplied with a temperature- dependent function (T) (Tucker & Koren 1971),
ERðsÞ ¼ n^2 e ðsÞ½TðsÞ ; ð 12 Þ
which was approximated by Rosner et al. (1978) by piece- wise power laws [see Appendix A in Rosner et al. 1978 for the definition of ðTÞ]. For a discussion of other calcula- tions of the radiative loss function and consequences on the hydrostatic solutions see x 4.5. The one-dimensional parameterization of loops with a distance coordinate s involves an angle ðsÞ between the magnetic field line (defining a loop) and the radial direction r. The simplest geometry employs semicircular loops, for which the height hðsÞ in the loop plane relates to the loop distance s by
hðsÞ ¼ rðsÞ R ¼
2 L
sin
s 2 L
; ð 13 Þ
with L the loop half-length. The derivative ðdh=dsÞ defines then the cosine of the angle used in the momentum bal- ance equation (2),
dr ds
dh ds
¼ cos
s 2 L
¼ cos : ð 14 Þ
For the variation of the loop cross section AðsÞ along the loop coordinate s we follow the line-dipole model of Vesecky, Antiochos, & Underwood (1979). In their model the inner and outer field line of a loop intersect in the lowest subphotospheric point, where the line dipole is buried, while the cross section varies as a sin^2 function, expanding by a factor of C from the photosphere to the loop apex. Because we are using a semicircular geometry for the loops, only loops with an expansion factor of ¼ 2 can be accommo- dated in the geometry of Vesecky et al. (1979). To allow for an arbitrary large range of expansion factors C in semicircu- lar loops we generalize the model of Vesecky et al. (1979) by relaxing the condition of a zero cross section at the sub- photospheric anti-apex point (s ¼ L). We define a general- ized cross section function
AðsÞ ¼ A 0 sin^2
s þ ssub L þ ssub
; ð 15 Þ
where the zero cross section point is located at position
270 ASCHWANDEN & SCHRIJVER Vol. 142
with the normalized space coordinate z ¼ s 0 þ ðs s 0 Þ=L and ¼ 0 :5. Both analytical solutions are shown in Figure 2, agreeing on the d5% level for most parts of the loop, after the pressure in the RTV model has been adjusted by a factor of p 0 0 :2985. The authors of the MKB model (Martens et al. 2000) note that they have not been able to reproduce the different analytical solution (eq. [17]) given by Rosner et al. (1978), derived under the same assumptions. No analytical expression or solution has been found for the general case of hydrostatic pressure equilibrium with nonuniform heating. In previous work (Aschwanden, Schrijver, & Alexander 2000b, 2001) we used for an approx- imation a ‘‘ generalized ’’ elliptical function, xa^ þ ya^ ¼ 1, where the variables x ¼ ðT T 0 Þ=ðTmax T 0 Þ represent a normalized temperature variable and y ¼ ðL sÞ=ðL s 0 Þ a normalized length variable. We found rough agreement with the exact hydrostatic solution TðsÞ within an accuracy level of a few percent (Fig. 2 in Aschwanden et al. 2000b). However, this approximation is not sufficient to derive other quantities in scaling laws, because the conductive flux requires the second-order derivative and multiplies uncer- tainties in the temperature function with a power of T^3 :^5
(eq. [10]). We therefore searched for a better approximation and found an extremely well-matching function by employ- ing independent power indices in the ‘‘ generalized ’’ ellipti- cal function, i.e., xa^ þ y^1 =b^ ¼ 1. Explicitly, we parameterize the temperature function with the following approximation:
TðsÞ ¼ Tmax 1
L s L s 0
ab : ð 19 Þ
We demonstrate the usefulness of this empirically found parameterization in Figure 2, by fitting it to the numerically obtained solution for a particular loop (Tmax ¼ 3 MK, L ¼ 100 Mm) with uniform heating sH 4 L. The difference between the numerical and analytical best fit (obtained for a ¼ 2 :012 and b ¼ 0 :3215) amounts to d 10 ^3 over the entire range of s 0 < s < L. The mean and standard devia- tion are ½TanaðsÞ TnumðsÞ=Tmax ¼ 0 : 0006 0 :0037. The differences to the analytical solutions of Rosner et al. (1978) and Martens et al. (2000) are within the d5% level, after adjusting the pressure by a factor of p 0 0 :2985 in the RTV model. Part of the discrepancy between the analytical approximations and numerical solution result from the
Temperature
0 10 20 30 40
T(s) [MK]
Tmax1= 1.000 MK Tmin1= 0.020 MK Lloop= 40.000 Mm sheat=4000.000 Mm
log[Temperature]
104 105 106 107 108 109 1010
104
105
106
107
T(s) [MK]
Conductive flux
0 10 20 30 40
-1.4•10^5
-1.2•10^5
-1.0•10^5
-8.0•10^4
-6.0•10^4
-4.0•10^4
-2.0•10^4
0
2.0•10^4
FC
(s)
Density
0 10 20 30 40
108
109
1010
1011
n(s) [cm
-3]
n 0 = 1.64e+10 cm-
log[Density]
104 105 106 107 108 109 1010
108
109
1010
1011
n(s)
Momentum balance
0 10 20 30 40
-1•10-
-5•10-
0
5•10-
1•10-
dp/ds [dyne cm
-3 ]
-dp/ds dpgrav/ds dptot/ds
Pressure
0 10 20 30 40 Loop distance s[Mm]
p(s) [dyne cm
-2]
p^0 = 9.03e-02 cm
log[Pressure]
104 105 106 107 108 109 1010 Loop distance log(s-s 0 )
p(s)
Energy balance
0 10 20 30 40 Loop distance s[Mm]
-6•10-
-4•10-
-2•10-
0
2•10-
4•10-
6•10-
dE(s)/dV*dt [erg cm
-^
-1s ]^ E_H0 = 4.096e-05 erg cm-3^ s- Eheat Etot
Erad
-Econd
Fig. 1.—Hydrostatic solution of a uniformly heated loop with a loop-top temperature of Tmax ¼ 1 :0 MK and loop half-length of L ¼ 40 Mm is shown, computed with the numeric code described in x 3. Note that the transition region is adequately resolved with the numeric code and the boundary conditions of Tðs 0 Þ ¼ 2 104 MK and vanishing flux dT=dsðs 0 Þ ¼ 0 are accurately met (see temperature profile in middle top panel ). The correctness of the numeric solution is warranted by the criteria of zero momentum along the loop (thick solid line in second right panel ) and zero energy balance along the loop (thick solid line in bottom right panel ).
272 ASCHWANDEN & SCHRIJVER Vol. 142
assumption of constant pressure in the RTV and MKB model, while the remaining discrepancy is attributed to dif- ferent approximations in the radiative loss function, as dis- cussed below. We are motivated to employ the same formalism for a larger parameter space, since our Ansatz of the temperature parameterization (eq. [19]) reproduces the numerical solution with extremely high accuracy (d 10 ^3 ) and has a simpler analytical form than the formulations given in equations (17) and (18). We fitted the analytical expression TðsÞ (eq. [17]) with the three free variables a, b, s 0 to all of our over 1000 numerical solutions TnumðsÞ in the entire parameter space of L, sH , Tmax and found that the power indices a and b essentially depend only on a single parameter, the ratio L=sH , but have no dependence on the maximum temperature Tmax or the parameters L and sH separately. In other words, the solu- tion of the temperature function is invariant in TðsÞ=Tmax and s=L. The proportionality of TðsÞ=Tmax is also evident in the analytical solutions of Rosner et al. (eq. [17]) and Martens et al. (eq. [18]). We found that the dependence of the temperature power indices aðL=sH Þ and bðL=sH Þ can best be fitted with the empirical functions:
aðL; SH Þ ¼ a 0 þ a 1
L
sH
a 2 ; ð 20 Þ
bðL; SH Þ ¼ b 0 þ b 1
L
sH
b 2 : ð 21 Þ
The best fits are shown in Figure 3, for the subset of hydro- static solutions with a maximum temperature of Tmax ¼ 3 MK, sorted by the parameter L=sH. The best-fit coefficients are given in Table 1. The similarity of the coefficients con- firms that there is no significant dependence on the loop-top temperature Tmax, and thus a and b are independent of Tmax. We run our analytical approximation of the temperature function (eqs. [19]–[21]) through all 1000 numerical solu-
tions with the same coefficients and find a relative accuracy of ½TðsÞ=TnumðsÞ 1 d 10 ^2 10 ^3. Thus, the parameteriza- tion of TðsÞ given with equations (19)–(21) provides us a simple analytical formulation of the temperature function that is sufficiently accurate in the entire parameter space and can be used as a powerful tool to solve the hydrostatic equations.
4.2. The Pressure Function pðsÞ After we have obtained a suitable approximation of the temperature function TðsÞ, we have a much easier way to determine the pressure function pðsÞ, because we can directly integrate the momentum equation (eq. [2]), where the density nðsÞ can be substituted by the ideal gas law nðsÞ ¼ pðsÞ= 2 kBTðsÞ (eq. [5]), so that the momentum equa- tion contains only the unknown pressure function pðsÞ,
dpðsÞ dh
pðsÞ 0
106 K
TðsÞ
1 þ
hðsÞ R
ð 22 Þ
0 20 40 60 80 100 Loop length coordinate s[Mm]
Temperature T[MK]
AS (numerical) AS (analytical) RTV (analytical) MKB (analytical)
L = 100 Mm Tmax = 3 MK uniform heating
Fig. 2.—Hydrostatic solution of a uniformly heated loop with a loop- top temperature of Tmax ¼ 3 MK and a half-length of L ¼ 100 Mm are shown (top panel ) from our numerical code (crosses), the analytical solution of Rosner, Tucker, & Vaiana (RTV: dashed line), the analytical solution of Martens, Kankelborg, & Berger (MKB; dashed-dotted line), and the best fit with our analytical function (AS), TðsÞ ¼ Tmaxf 1 ½ðL sÞ=ðL s 0 Þ^2 :^0074 g^0 :^2858 (solid line). The base pres- sure in the RTV model had to be adjusted by a factor of pRTV 0 ¼ p 0 0 :2985 to match our numeric solution.
0.01 0.10 1.00 10.0 0 L/s_H
a(L/s
)H
a0= 2. a1= 0. a2= 1. a =a0+a1*(L/s_h)a dT<0. N = 262
0.01 0.10 1.00 10.0 0 L/s_H
b(L/s
)H (^) b0= 0. b1= -0. b2= 0. b =b0+b1*(L/s_h)b dT<0. N = 262
0.01 0.10 1.00 10.0 0 L/s_H
qλ
(L/s
)H
c0= 0. c1= 0. c2= 1. q =c0+c1*(L/s_h)c dT<0. N = 262
Fig. 3.—Fits of the power indices aðL=sH Þ (eq. [20]; top panel ) and bðL=sH Þ (eq. [21]; middle panel ) in the temperature function TðsÞ (eq. [19]), and for the scale height factor qðL=sH Þ (eq. [27]; bottom panel ) in the pres- sure function pðL=sH Þ (eqs. [25] and [26]), shown for 262 numeric solutions with Tmax ¼ 3 MK.
No. 2, 2002 HYDROSTATIC APPROXIMATIONS OF CORONAL LOOPS 273
(e.g., p 0 and EH 0 ) on the three independent parameters is specified by two scaling laws, i.e., p 0 (L, sH , Tmax) and EH 0 (L, sH , Tmax). These two scaling laws have been derived in Rosner et al. (1978) and Serio et al. (1981) by integrating the energy equation in two different ways: (1) as spatial integral
R
f ðsÞds and (2) as temperature inte- gral
R
f ðTÞdT after substituting the conductive flux vari- able FC ðTÞ. Rosner et al. (1978) derived the scaling laws under the fol- lowing assumptions and approximations:
- Constant pressure, pðsÞ ¼ p 0.
- Uniform heating, sH ¼ 1.
- Radiative loss function is approximated with single power law, ðTÞ 0 T^1 =^2 , with 0 ¼ 10 ^18 :^81 erg cm^3 s^1.
R 4. Auxiliary^ function^ fH^ ðTÞ^5 fRðTÞ,^ i.e., T^5 =^2 EH ðTÞdT 5
R
T^5 =^2 ERðTÞdT.
- Neglect height dependence of solar gravitation gðhÞ ¼ gð 1 þ h=RÞ^2 (eq. [9]).
- Footpoint in photosphere (s 0 ¼ 0).
The scaling laws can generally be expressed as a function of the independent variables [L 0 , sH , Tmax] by
p 0 ðL 0 ; sH ; TmaxÞ ¼
L 0
Tmax S 1
; ð 29 Þ
EH 0 ðL 0 ; sH ; TmaxÞ ¼ L 0 2 Tmax^7 =^2 S 2 : ð 30 Þ
where we denoted the footpoint-apex distance by L 0 ¼ L s 0. Under the assumptions and approximations listed above, Rosner et al. (1978) derived the following constants for the expressions S 1 and S 2 :
SRTV 1 ¼ 1400 ; ð 31 Þ
SRTV 2 ¼ 0 : 95 10 ^6 : ð 32 Þ
We show the comparison of the RTV scaling laws with our numerical solutions as a function of L=sH in Figure 4 (top panels). The scaling law for the base pressure agrees with the numerical solutions within ðpRTV 0 =p 0 Þ 0 : 9 0 :1 for near- uniform heating (L=sH d1). The scaling law for the base heating rate agrees with the numerical solutions within ðE HRTV 0 =EH 0 Þ 0 : 8 0 : 3 for near-uniform heating (L=sH d 1 Þ but yields too low heating rates down to frac- tions of 0.2 for short heating scale lengths (at L=sH d2). Serio et al. (1981) generalized the RTV scaling laws for variable pressure (owing to gravity) and nonuniform heat- ing but retained the other approximations from the deriva- tion of Rosner et al. (1978). Thus, Serio’s derivation is subject to the same set of assumptions and approximations except the first two:
- The pressure is assumed to be an exponential function of the loop length, pðsÞ p 0 expðs=spÞ, with sp ¼ 0 TMK. This approximation neglects the temperature variation TðsÞ along the loop, and thus the variation of the pressure scale height pðsÞ (see eq. [26]).
- The height dependence in the pressure function is approximated by the semicircular loop coordinate s, pðhÞ pðsÞ.
Note that the numerical calculations of coefficients , , (^0) , (^0) (eqs. [3.7]–[3.8] in Serio et al. 1981) are optimized
based on numerical solutions in some (unspecified) parame- ter space, which probably covers a different parameter
regime than our numerical solutions. Moreover, Serio et al. calculate hydrostatic solutions for loops with an expansion factor of 5, while we calculate cases for constant as well as expanding cross sections separately. Serio’s scaling laws have the same basic dependence on L 0 and Tmax as the RTV laws (eqs. [29]–[30]) but differ in the scaling law expressions S 1 and S 2 (eqs. [31]–[32]),
S 1 Serio ¼ 1400 exp 0 : 08
L 0
sH
L 0
sp
; ð 33 Þ
SSerio 2 ¼ 0 : 95 10 ^6 exp 0 : 78
L 0
sH
L 0
sp
; ð 34 Þ
where we denoted sp ¼ 0 TMK and TMK ¼ Tmax= 106 MK. While Serio’s generalization accounts for nonuniform heat- ing and pressure variation, the approximations made in the derivation lead to differences from the proper numerical sol- utions, as shown in Figure 4 (middle row) relative to our exact numerical solutions. The agreement with our numeri- cal solutions of p 0 and EH 0 are within 1: 0 dpSerio 0 =p 0 d 1 : 4 and 0: 9 dE HSerio 0 =EH 0 d 1 :3, with some extreme deviations down to E HSerio 0 =EH 0 e 0 :2. In order to achieve a higher level of accuracy between the numerical hydrostatic solutions and the scaling law approx- imations, we add two additional correction terms to Serio’s expressions (eqs. [33] and [34]), leading to five coefficients for each of the two scaling laws, called di and ei, i ¼ 0 ;... ; 4, respectively:
SAS 1 ¼ d 0 exp d 1
L 0
sH
þ d 2
L 0
sp
þ d 3
L 0
sH
þ d 4
L 0
sp
; ð 35 Þ
SAS 2 ¼ e 0 exp e 1
L 0
sH
þ e 2
L 0
sp
þ e 3
L 0
sH
þ e 4
L 0
sp
: ð 36 Þ
We determine these 10 coefficients di and ei by minimizing the differences of the scaling law expressions p 0 ðdiÞ (eqs. [29] and [35]) [and EH 0 ðeiÞ (eqs. [30] and [36])] to the numerical solutions pnum 0 [and Enum H 0 ] from our 1000 numerical runs, which cover the parameter space of [Tmax ¼ 1 10 MK, L ¼ 4 400 Mm, sH ¼ 4 400 Mm]. The best-fit values of the coefficients di and ei, i ¼ 0 ;... ; 4 are tabulated in Table 1, for different temperatures Tmax ¼ 1, 3, 5, and 10 MK. If one uses just one set of coefficients (say from Tmax ¼ 3 :0 MK in Table 1) for a larger temperature range, the accuracy of the scaling laws is about d5% in the temperature of T ¼ 2 5 MK and degrades to 10%–20% in the temperature range of T ¼ 1 10 MK. For a higher accuracy in the order of a few percent, a spline interpolation of the coefficients (given in Table 1) as a function of Tmax is recommended. These empirical scaling laws (eqs. [35] and [36]) provide a best fit to the numerical solutions within an accuracy of a few per- cent (see Fig. 4, bottom panels). The functional dependence of these scaling laws is shown in Figure 5 for EH 0 (L, sH , Tmax), and in Figure 6 for p 0 (L, sH , Tmax), respectively.
4.4. Choice of Independent Parameters ½L; sH ; EH 0 The analytical formulation of the scaling law derived by Serio et al. (1981) requires the independent parameter set [L; sH ; Tmax], because the pressure scale height sp ¼ 0 ðTmax= 106 K) depends on Tmax, so that the second scaling law (eqs. [30] and [34]) can only be expressed explic- itly for EH 0 ðL; sH ; TmaxÞ, but not explicitly in the form of TmaxðL; sH ; EH 0 Þ. The same is also true for our modified
No. 2, 2002 HYDROSTATIC APPROXIMATIONS OF CORONAL LOOPS 275
scaling law (eqs. [30] and [36]). However, from a physical point of view, the choice of the independent parameter set ½L; sH ; EH 0 is more natural, because the heating function specified by ½sH ; EH 0 defines the energy input, while the temperature Tmax represents the final outcome of a relaxa- tion process when evolving into a hydrostatic equilibrium.
This is especially important for hydrodynamic processes, where the input can be defined as initial boundary condi- tion, while the parameters of the final outcome generally cannot be predicted. Only for hydrostatic solutions, the choice of independent parameters are exchangeable in the scaling law relations.
RTV scaling law
L/s_H
E
H
RTV
/E
H
(^) TTmaxmax= 1 MK, q= 1 MK, qE_H0E_H0= 1.06+ 0.31= 1.06_
RTV scaling law
L/s_H
p^0
RTV
/p
0
TTmaxmax= 1 MK, q= 1 MK, qp_0p_0= 0.95+ 0.14= 0.95_
Serio scaling law
L/s_H
E
H
Serio
/E
H
T
Tmaxmax= 1 MK, q= 1 MK, qE_H0E_H0= 1.16+ 0.27= 1.16_
Serio scaling law
L/s_H
p^0
Serio
/p
0 TTmaxmax= 1 MK, q= 1 MK, qp_0p_0= 1.23+ 0.17= 1.23_
Aschwanden + Schrijver
L/s_H
E
H
AS
/E
H
TTmaxmax= 1 MK, q= 1 MK, qE_H0E_H0= 1.00+ 0.08= 1.00_
Aschwanden + Schrijver
L/s_H
p
AS 0
/p
0
TTmaxmax= 1 MK, q= 1 MK, qp_0p_0= 1.00+ 0.02= 1.00_
RTV scaling law
L/s_H
E
H
RTV
/E
H
TTmaxmax= 3 MK, q= 3 MK, qE_H0E_H0= 0.85+ 0.25= 0.85_
RTV scaling law
L/s_H
p^0
RTV
/p
0
TTmaxmax= 3 MK, q= 3 MK, qp_0p_0= 0.90+ 0.12= 0.90_
Serio scaling law
L/s_H
E
H
Serio
/E
H
TTmaxmax= 3 MK, q= 3 MK, qE_H0E_H0= 1.10+ 0.05= 1.10_
Serio scaling law
L/s_H
p^0
Serio
/p
0
TTmaxmax= 3 MK, q= 3 MK, qp_0p_0= 1.06+ 0.07= 1.06_
Aschwanden + Schrijver
L/s_H
E
H
AS
/E
H
TTmaxmax= 3 MK, q= 3 MK, qE_H0E_H0= 1.00+ 0.03= 1.00_
Aschwanden + Schrijver
L/s_H
p
AS 0
/p
0
TTmaxmax= 3 MK, q= 3 MK, qp_0p_0= 1.00+ 0.02= 1.00_
RTV scaling law
L/s_H
E
H
RTV
/E
H
TTmaxmax= 5 MK, q= 5 MK, qE_H0E_H0= 0.86+ 0.26= 0.86_
RTV scaling law
L/s_H
p^0
RTV
/p
0
TTmaxmax= 5 MK, q= 5 MK, qp_0p_0= 0.93+ 0.12= 0.93_
Serio scaling law
L/s_H
E
H
Serio
/E
H
TTmaxmax= 5 MK, q= 5 MK, qE_H0E_H0= 1.16+ 0.03= 1.16_
Serio scaling law
L/s_H
p^0
Serio
/p
0
TTmaxmax= 5 MK, q= 5 MK, qp_0p_0= 1.08+ 0.04= 1.08_
Aschwanden + Schrijver
L/s_H
E
H
AS
/E
H
TTmaxmax= 5 MK, q= 5 MK, qE_H0E_H0= 1.00+ 0.02= 1.00_
Aschwanden + Schrijver
L/s_H
p
AS 0
/p
0
TTmaxmax= 5 MK, q= 5 MK, qp_0p_0= 1.00+ 0.01= 1.00_
RTV scaling law
L/s_H
E
H
RTV
/E
H
TTmaxmax=10 MK, q=10 MK, qE_H0E_H0= 0.97+ 0.30= 0.97_
RTV scaling law
L/s_H
p^0
RTV
/p
0
TTmaxmax=10 MK, q=10 MK, qp_0p_0= 1.13+ 0.10= 1.13_
Serio scaling law
L/s_H
E
H
Serio
/E
H
TTmaxmax=10 MK, q=10 MK, qE_H0E_H0= 1.39+ 0.06= 1.39_
Serio scaling law
L/s_H
p^0
Serio
/p
0
TTmaxmax=10 MK, q=10 MK, qp_0p_0= 1.31+ 0.07= 1.31_
Aschwanden + Schrijver
L/s_H
E
H
AS
/E
H
TTmaxmax=10 MK, q=10 MK, qE_H0E_H0= 1.00+ 0.01= 1.00_
Aschwanden + Schrijver
L/s_H
p
AS 0
/p
0
TTmaxmax=10 MK, q=10 MK, qp_0p_0= 1.00+ 0.00= 1.00_
Fig. 4.—Accuracy of the two scaling laws p 0 ðL; sH ; TmaxÞ (left column) and EH 0 ðL; sH ; TmaxÞ (right column) is shown as a function of the parameter ðL=sH Þ, for the Rosner-Tucker-Vaiana scaling law for uniform heating (top), for Serio’s scaling law for nonuniform heating (middle), and for the analytical approxima- tions in this study (bottom). All ratios are normalized by the values of the proper numerical solutions for p 0 and EH 0. In each panel we show the four subsets for different temperatures (Tmax ¼ 1, 3, 5, 10 MK) with separate symbols, and the averages and standard deviations of the ratios are given for each temperature separately.
276 ASCHWANDEN & SCHRIJVER Vol. 142
dances yields a somewhat higher temperature function and a massively enhanced density and pressure function (Fig. 8, black curves), while the solution based on coronal abundan- ces yields a lower density and pressure (Fig. 8, gray curves), due to the enhanced radiative losses of the iron element at temperatures around T 1 MK. Thus, this uncertainty by about a factor of 2 in the assumptions on elemental abun- dances far outweighs the inaccuracy of our analytical approximation to the numerical hydrostatic solutions within the few percent level.
- GENERALIZED HYDROSTATIC SOLUTIONS In this section we generalize our analytical approxima- tions of the hydrostatic solutions to allow applications in a wider range of observational circumstances. We generalize the solutions for extremely small heating scale lengths, for inclined loops, for loops with variable cross sections, and for slow velocity flows. These generalizations have been tested in part of the previously used parameter space, which covers a temperature range of T ¼ 1 10 MK.
5.1. Small Heating Scale Heights In the previous sections we covered a large parameter space with spatial scales in the range of 4–400 Mm for L and sH , but we excluded extremely small heating scale lengths, say with sH dL=3. Already Serio et al. (1981) subdivided hydrostatic solutions into two classes with the same crite- rion: class I are loops with the temperature maximum at the loop top (which is the case for sH eL=3), and class II are loops with the temperature maximum at some intermediate position between the loop top and footpoints (which is the case for sH dL= 3 Þ. We show numerical solutions of hydro- static temperature profiles from sH ¼ L down to sH ¼ L= 25 in Figure 9 (solid lines). The temperature maximum clearly moves downward the loop with decreasing heating scale length ratio sH =L, and a larger segment of the loop becomes near isothermal. Because our previous temperature approx- imation with a generalized ellipse function xa^ þ x^1 =b^ ¼ 1 has its maximum by definition at the loop top, the same approx- imation cannot represent loops with an intermediate tem- perature maximum. However, a correction can be added that makes the analytical approximation valid down to extremely short heating scale lengths of sH eL=25. Because the temperature solution was found to be nearly invariant with respect to the normalized temperature TðsÞ=Tmax and spatial coordinate z ¼ ðs s 0 Þ=ðL s 0 Þ, the correction term scales only with the ratio sH =L. We found a good approxi- mation within the d1% level (Fig. 9, dashed lines) with the
Temperature
T(s) [MK]
Tmax= 1 MK Lloop= 40 Mm sheat=317 Mm
Density
5.0•10^8
1.0•10^9
1.5•10^9
2.0•10^9
n(s) [cm
]
Pressure
p(s) [dyne cm
]
Chromospheric abundances (Meyer)
Coronal abundances (Feldman)
Conductive flux
-1.2•10^5
-1.0•10^5
-8.0•10^4
-6.0•10^4
-4.0•10^4
-2.0•10^4
2.0•10^4
F
(s)C
Fig. 8.—Comparison of hydrostatic solutions computed for two differ- ent radiative loss functions: for the RTV six–power-law approximation (Fig. 7, thick line) and chromospheric abundances according to Meyer (1985), and for the same radiative loss function and coronal abundances according to Feldman (1992). The difference in the solution is shown in gray.
0.0 0.2 0.4 0.6 0.8 1. 0 Loop length normalized (s-s 0 )/(L-s 0 )
Temperature normalized T/T
max
sH/L= 1.
sH/L= 0.
sH/L= 0.
sH/L= 0.
sH/L= 0.
sH/L= 0.
sH/L= 0.
sH/L= 0.
sH/L= 0.
sH/L= 0.
sH/L= 0.
sH/L= 0.
sH/L= 0.
sH/L= 0.
Heating scale height sH = 4 Mm Loop lengths L = 4-100 Mm
Fig. 9.—Hydrostatic solutions for extremely short heating scale lengths, sH =L ¼ 0 : 04 ;... ; 1 :0. The numeric solutions are shown in solid lines, and the analytical approximation (eqs. [40] and [41]) in dashed lines.
278 ASCHWANDEN & SCHRIJVER Vol. 142
following corrected temperature function:
z ¼
L s L s 0
; ð 38 Þ
T^0 ðsÞ ¼ Tmax½ 1 zab^1 þ 0 : 510 log
L
sH
ð 1 zÞz^5
: ð 39 Þ
Because our analytical approximations of the hydrostatic solutions and scaling laws are all expressed in terms of the temperature solution (Table 2), the standard temperature approximation TðsÞ can simply be replaced by the corrected function T^0 ðsÞ in this formalism, and the analytical approxi- mations for the pressure function pðsÞ and density nðsÞ as well as the resulting scaling laws will automatically be cor- rected as a function of the improved temperature function T^0 ðsÞ. This correction, however, needs only to be applied for
extremely short heating scale lengths, in the range of sH < L=3.
5.2. Inclined Loops Most of the observed coronal loops have some inclination of the average loop plane with respect to the vertical on the solar surface. For instance, a bundle of 30 stereoscopically reconstructed active region loops were found to have an almost uniform distribution of inclination angles in the range of ¼ 49 ^... þ 69 ^ (Aschwanden et al. 1999). While the gravitational scale height is strictly measured in vertical direction, the effective scale height in the loop plane varies with the cosine of the vertical scale height, so we can define an effective gravity component along the loop,
geff ¼ g cos : ð 40 Þ
TABLE 2 Summary of Analytical Formulae to Calculate Hydrostatic Solutions and Scaling Laws
Description Formula
Constants: Height of loop base......................... s 0 ¼ 1 : 3 108 cm Temperature at loop base ............... T 0 ¼ 2 : 0 104 K Solar radius .................................... R ¼ 6 : 96 1010 cm Solar gravity................................... g ¼ 2 : 74 104 cm s^2 Spitzer conductivity........................ ¼ 9 : 2 10 ^7 erg s^1 cm^1 K7/
Independent variables: Loop half-length ............................ L (cm) Heating scale length ....................... sH (cm) Loop-top temperature.................... Tmax (K) Base heating rate ............................ EH 0 (ergs cm^3 s^1 ) Loop plane inclination angle .......... h (deg) Loop expansion factor ................... C 1 Choice 1: [L, sH , Tmax, h, C]............. Choice 2: [L, sH , EH 0 , h, C].............. CTmax 55 : 2 E H^0 :^9770 L^20 exp
0 : 687 ðL 0 =s H Þ
(^2) = 7
Dependent parameters: Half-loop length above base ........... L 0 ¼ L s 0 Loop height.................................... h 1 ¼ ð 2 L=Þ Subphotospheric zero point............ ssub ¼ L ½ð = 2 Þ= arcsinð 1 =^1 =^2 Þ 1 ^1 Equivalent heating scale length....... s H ¼ sH ½ 1 þ ð 1 ÞðsH =LÞ^1 =^2 if (sH L, C 1) s H ¼ sH ½ð 1 Þ^1 =^2 if (sH > L, C > 1) Temperature index 1....................... a ¼ a 0 þ a 1 L 0 =s H
a 2
Temperature index 2....................... b ¼ b 0 þ b 1 L 0 =s H
b 2
Scale height factor .......................... q ¼ c 0 þ c 1 L 0 =s H
c 2
Effective gravity.............................. geff ¼ g cos Effective scale height....................... 0 ¼ 2 kB 106 ½K=lmpgeff
¼ 4 : 6 109 ð 1 = cosÞ cm Serio scale height ............................ sp ¼ 0 ðT (^) max= 106 KÞ Scaling law factor 1 ........................ S 1 ¼ d 0 exp
d 1 ðL 0 =s H Þ þ d 2 ðL 0 =spÞ
þ d 3 ðL 0 =s H Þ þ d 4 ðL 0 =spÞ Scaling law factor 2 ........................ S 2 ¼ e 0 exp
e 1 ðL 0 =s H Þ þ e 2 ðL 0 =spÞ
þ e 3 ðL 0 =s H Þ þ e 4 ðL 0 =spÞ Base heating rate (for Choice 1) ...... EH 0 ¼ L 0 2 Tmax^7 =^2 S 2 Base pressure.................................. p 0 ¼ L 0 1 ðTmax=S 1 Þ^3
Analytical approximations: Normalized length coordinate ........ zðsÞ ¼ ðL sÞ=ðL s 0 Þ Height (in loop plane)..................... h^0 ðsÞ ¼ h 1 sin ðs =h 1 Þ Loop cross section area .................. AðsÞ ¼ sin^2
ð= 2 Þðs þ ssubÞ=ðL þ ssubÞ
Temperature (if s H =L > 0 :3) .......... TðsÞ ¼ Tmax ½ 1 zab Temperature (if s H =L 0 :3) .......... TðsÞ ¼ Tmax½ 1 zab^ ½ 1 þ 0 : 510 log ðL =sHÞð 1 zÞz^5 Conductive flux .............................. FC ðsÞ ¼ TðsÞ^5 =^2 ½dTðsÞ=ds Pressure scale height....................... pðsÞ ¼ 0 ½TðsÞ= 106 K½ 1 þ h^0 ðsÞ=Rq Pressure.......................................... pðsÞ ¼ p 0 expf½h^0 ðsÞ h^0 ðs 0 Þ=pðsÞg Density........................................... nðsÞ ¼ ½pðsÞ= 2 kBTðsÞ
No. 2, 2002 HYDROSTATIC APPROXIMATIONS OF CORONAL LOOPS 279
With this concept we have a simple correction formula for the hydrostatic solutions of loops with expanding loop cross sections. Essentially, we can use the same hydrostatic solu- tions of loops with constant cross sections, say for a param- eter set of [EH 0 , s H , L, ¼ 1], to obtain a good approximation for a loop with expansion factor C. We dem- onstrate this in Figure 10, where the corresponding approxi- mations (dashed lines) are shown along with the exact numerical solutions (solid lines). The approximations match the exact numerical solutions with an accuracy of a few per- cent for ¼ 1 10 in coronal heights. So, the expansion fac- tor C of loops has an effect similar to that of a shorter heating scale length, i.e., the temperature profile is more iso- thermal in the upper part of the loops and the base pressure increases, compared with a loop with constant cross section. For very high expansion factors C, the temperature maxi- mum moves from the loop top downward, similar to the case of shorter heating scale lengths, sH 5 L.
5.4. Differential Emission Measure Modeling For modeling of stellar atmospheres one can use mag- netic field models and populate individual magnetic field lines with hydrostatic flux tubes. The total differential emis- sion measure (DEM) distribution from a stellar atmosphere can then be computed by adding up all (spatially unre- solved) flux tubes, yielding a global DEM distribution dEMðTÞ=dT (e.g., Schrijver & Aschwanden 2002; Peres et al. 2000, 2001). The DEM distribution of a single loop is defined by
dEMðTÞ dT
¼ AðTÞn^2 e ðs½TÞ
ds½T dT
; ð 49 Þ
which can be calculated by inverting the temperature profile TðsÞ and using the density function neðsÞ from our hydro- static solutions. One question is how accurate our analytical solutions match the numerical solutions of such DEM dis- tributions. We calculated these DEM distributions for both our numerical solutions dEMðTÞ=dTnum as well as for our analytical approximations dEMðTÞ=dTana and tested the accuracy of the mean ratio. Figure 11 shows a representative set of DEMs, calculated for four different temperatures (Tmax ¼ 1, 3, 5, 10 MK) and four different loop expansion factors ( ¼ 1, 3, 5, 10), for a loop with a length of L ¼ 40 Mm and a heating scale height of sH ¼ 20 Mm (Fig. 11, left panel) as well as for uniform heating, sH 4 L (Fig. 11, right panel). The average deviations between the numerical solu- tions and analytical approximations are found of the order of 5%–10% over the logarithmically spaced temperature range of T > 0 :5 MK.
5.5. Loops with Subsonic Flows The hydrostatic solutions calculated here represent solu- tions of the general magnetohydrodynamic equations (eqs. [1]–[3]) in the limit of no flows, vðsÞ ¼ 0. The solutions for small flow velocities vðsÞ (compared with the sound speed, vðsÞ 5 cs), however, are expected to be still close to this asymptotic limit of the hydrostatic case. For slow flows we can therefore take the solutions of the temperature profile TeðsÞ and density profile neðsÞ we obtained in the hydrostatic limit, and insert them into the equation of mass conserva-
tion (eq. [1]) to obtain the velocity profile vðsÞ,
vðsÞ v 0
n 0 A 0 nðsÞAðsÞ
v 0
T 0 A 0
TðsÞAðsÞ
exp
h^0 ðsÞ h^0 ðs 0 Þ pðsÞ
ð 50 Þ
This equation tells us that the flow speed has the tendency to increase with height to balance the mass in an exponentially dropping density atmosphere, at least for loops with con- stant cross section. If the loops are significantly expanding with height, the tendency for accelerated flows goes away. For instance, if the loop expansion is about a factor 2.7 per scale height, the area expansion compensates exactly for the e-folding density decrease and siphon flows can exist almost with a constant speed. The sound speed for Te ¼ 1 :0 MK is cs ¼ 111 ðTMKÞ^1 =^2 km s^1 ¼ 111 km s^1 , with ¼ 1 for an ideal gas (or ¼ 5 =3 for an adiabatic gas, respectively). We expect therefore that loops with moderate upflow speeds, of say v 0 d10 km s^1 , still fulfil the criterion v 05 cs, and thus can be accurately characterized by our analytical approximations of the hydrostatic solutions.
- DISCUSSION We have derived accurate analytical approximations of the hydrostatic solutions under the following assumptions: (1) time independent, (2) no flows, and (3) semicircular geometry. The solutions are laid out for uniform or nonuni- form heating functions (characterized by an exponential scale length) and account for the vertical variation of solar gravity, for inclined loop planes, and for variable cross sec- tions. Although the real solar corona displays a variety of dynamic processes, hydrostatic solutions are still valid in first order, if mass flows and geometric changes are slow compared with the sound speed. Therefore, the analytic approximations derived here have still a quite accurate val- idity for coronal structures that have flows and are slowly evolving compared on a hydrodynamic timescale. Also we calculated general corrections for loops with expanding cross sections, which have been calculated previously only for a few special cases (Vesecky et al. 1979; Serio et al. 1981). In the following, we discuss some applications where the analytical approximations may be most useful, the main motivation of this analytical study:
- Faster convergence of numerical codes.—The previ- ously derived scaling laws for loop base pressure p 0 ðL; sH ; TmaxÞ and EH 0 ðL; sH ; TmaxÞ (Serio et al. 1981) involve approximations that cause deviations up to a factor of d5 from the exact numerical solutions, and thus render the convergence of numerical codes problematic in some parameter regimes, in particular for short heating scale lengths sH 5 L, where numerical convergence is difficult, but which seems to be the most realistic situation for coro- nal loops. The more accurate scaling laws derived here thus speed up numerical codes significantly in those regimes.
- Multithread models of coronal loops.—Coronal loops seem to consist of unresolved threads even when observed with the highest-resolution instruments down to 1^00. The loop threads are likely to be thermally insulated from each other and thus are likely to have independent temperatures. Previous physical modeling of active region loops has been done in the form of single–flux tube concepts (e.g., Landini & Monsignori-Fossi 1975; Craig, McClymont, & Under-
No. 2, 2002 HYDROSTATIC APPROXIMATIONS OF CORONAL LOOPS 281
wood 1978; Rosner et al. 1978; Serio et al. 1981), or in the form of multithread concepts (e.g., Reale & Peres 2000; Aschwanden et al. 2000b). The single–flux tube concept is appropriate for bright loops that can be clearly separated from the background and if they are observed with a nar- rowband filter, such as with the Extreme-ultraviolet Imag- ing Telescope (EIT/SOHO) (Neupert et al. 1998; Aschwanden et al. 1999, 2000a) or with the Transition Region and Coronal Explorer (TRACE) (Lenz et al. 1999; Aschwanden et al. 2000b). Single–flux tube modeling, how- ever, is less reliable for faint loops, because the background
subtraction (of the coronal background that may have dif- ferent temperature structure) has much larger uncertainties. Using the analytical approximations derived here, a multi- thread model of a coronal loop can be parameterized straightforward, using a distribution of base heating rates NðEH 0 Þ for an ensemble of loop threads, and the synthe- sized emission measure can be integrated and fitted to the data.
- Hydrostatic weighting bias.—Another severe disad- vantage of single–flux tube modeling is the hydrostatic weighting bias (Aschwanden & Nitta 2000), when broad-
106 107 Temperature T [K]
1020
1021
1022
1023
1024
1025
1026
Differential Emission Measure dEM/dT=A(T)n
(^2) e (T)(ds(T)/dT) [cm
-^
K
-1^ ]
ΓΓ= 1 : 1.10+0.04= 1 : 1.10_
ΓΓ= 2 : 1.03+0.05= 2 : 1.03_
ΓΓ= 5 : 0.91+0.06= 5 : 0.91_
ΓΓ=10 : 0.91+0.09=10 : 0.91_
Tmax = 1 MK :
ΓΓ= 1 : 0.94+0.02= 1 : 0.94_
ΓΓ= 2 : 0.86+0.03= 2 : 0.86_
ΓΓ= 5 : 0.82+0.05= 5 : 0.82_
ΓΓ=10 : 0.93+0.06=10 : 0.93_
Tmax = 3 MK :
ΓΓ= 1 : 0.93+0.03= 1 : 0.93_
ΓΓ= 2 : 0.87+0.05= 2 : 0.87_
ΓΓ= 5 : 0.89+0.06= 5 : 0.89_
ΓΓ=10 : 1.19+0.10=10 : 1.19_
Tmax = 5 MK :
ΓΓ= 1 : 0.99+0.03= 1 : 0.99_
ΓΓ= 2 : 0.97+0.04= 2 : 0.97_
ΓΓ= 5 : 0.93+0.07= 5 : 0.93_
ΓΓ=10 : 0.99+0.17=10 : 0.99_
Tmax =10 MK :
sH = 20 Mm L = 40 Mm Tmax = 1, 3, 5, 10 MK Γ = 1, 2, 5, 10
106 107 Temperature T [K]
1020
1021
1022
1023
1024
1025
1026
ΓΓ= 1 : 0.99+0.02= 1 : 0.99_
ΓΓ= 2 : 1.05+0.05= 2 : 1.05_
ΓΓ= 5 : 0.94+0.08= 5 : 0.94_
ΓΓ=10 : 0.90+0.10=10 : 0.90_
Tmax = 1 MK :
ΓΓ= 1 : 0.99+0.02= 1 : 0.99_
ΓΓ= 2 : 1.12+0.01= 2 : 1.12_
ΓΓ= 5 : 1.09+0.04= 5 : 1.09_
ΓΓ=10 : 1.13+0.08=10 : 1.13_
Tmax = 3 MK :
ΓΓ= 1 : 0.98+0.03= 1 : 0.98_
ΓΓ= 2 : 1.09+0.02= 2 : 1.09_
ΓΓ= 5 : 1.05+0.04= 5 : 1.05_
ΓΓ=10 : 1.07+0.08=10 : 1.07_
Tmax = 5 MK :
ΓΓ= 1 : 1.01+0.06= 1 : 1.01_
ΓΓ= 2 : 1.03+0.03= 2 : 1.03_
ΓΓ= 5 : 0.96+0.04= 5 : 0.96_
ΓΓ=10 : 0.93+0.07=10 : 0.93_
Tmax =10 MK :
uniform heating (sH >> L)
Fig. 11.—Accuracy of differential emission measure (DEM) distributions is shown by comparing the analytical approximations of dEMðTÞ=dTana (smooth curves) to the numerical solutions dEMðTÞ=dTnum (histograms), for a representative subset of numerical solutions in four temperature ranges, four loop expansion factors, and for two different heating scale lengths (i.e., sH ¼ 20 Mm in left panel, and uniform heating in right panel). Note that the average difference is of order 5%–10%.
282 ASCHWANDEN & SCHRIJVER Vol. 142
